It is known that almost every \(x\)  in the interval \(I_{\beta } = [0, \frac{1}{\beta - 1}]\)  has a continuum of different representation of the form (REF ) ([1]} and [2]}).
We will be working with a special form of the representation (REF ) that was introduced in [3]} under the name \(\beta \) -expansion of \(x\)  and is defined by the condition \(d_n = [\beta T_{\beta }^{n-1} (x)]\)  for \(n \ge 1\) .
